Download Computation of hadronic two-point functions in Lattice QCD

Computation of hadronic two-point functions in Lattice
QCD
Lecture by Gunnar Bali
Notes by Michèle Wandelt, Florian Gruber and Gunnar Bali
12.1.2012
1
Introduction
Strongly interacting particles are called hadrons. These are composed of quarks and gluons.
Their masses can be determined in Lattice simulations, from so-called two-point functions,
where the particle of interest is created at some initial time and destroyed at a later time, on
a four dimensional Euclidean spacetime lattice. The ground state mass is then obtained from
the exponential decay at large time differences. We provide a very brief introduction into
Lattice QCD and the notations used. Subsequently, the standard methods to construct and
to compute these two-point functions and hadron masses are introduced and explained for a
simple example. This follows on from the previous lecture where so-called quark propagators
are obtained by solving a sparse linear system. These are the building blocks of the two-point
functions discussed here.
Remark: we employ the notation As = b, instead of Ax = b, because in physics x often
refers to an element of R4 .
1
1.1
1.1.1
General overview
Particles in the standard model
The standard model of particle physics contains different elementary particles, which, in this
theory, have no substructure:
Figure 1: Elementary particle content of the standard model of particle physics (from
http://www.wikipedia.org).
• Leptons: electron e− , muon µ− , tau τ − with electric charges −e and the electrically
neutral neutrinos νe , νµ , ντ .
• Quarks come in 6 flavours: up, down, strange, charm, bottom (or beauty) and top.
They carry so-called colour charges and interact via the strong (= colour) interaction.
This force is described by Quantum Chromodynamics. In addition, quarks carry electric charges (u, c, t : + 23 e, d, s, b : − 13 e) and weak charges. They interact with other
fermions both electromagnetically and weakly. Each quark has a baryon number 1/3.
• Leptons and quarks are fermions. All known elementary fermions have spin 12 .
• The strong force is mediated by gauge bosons (spin 1 particles) called gluons. Gluons
are massles but, unlike the electrically neutral photon, carry (colour) charges.
• One important property of QCD is confinement: free “coloured” objects like quarks or
gluons cannot be observed in Nature. Due to the strong interaction, they are always
2
bound in hadrons. Hadrons can either be mesons (integer spin) or baryons (half-integer
spin).
• Mesons have baryon number zero, i.e. they contain as many quarks as antiquarks. In
the quark model a meson contains one quark and one antiquark (q̄q). QCD is more
complex and mesons are expected to contain higher “Fock states” like tetraquarks
(q̄q q̄q) as well. Some mesons may even contain glueball components (with no quarks).
¯ In Nature there are
The pion contains u and d quarks and their anti-particles ū and d.
+
−
¯
¯ π : ūd, π 0 : √1 (ūu − dd).
two electrically charged and one neutral pion: π : du,
2
• Baryons have baryon number one, i.e. they contain three valence quarks. (The number
of quarks minus the number of antiquarks equals three.) Within the quark model,
baryons contain only valence quarks and no antiquarks. The most prominent examples
are the proton (uud, charge +e) and the neutron (udd, electrical charge 0). Proton
and neutron are also called nucleons since they are the constituents of atomic nuclei.
There is particle-antiparticle creation in quantum field theories. This leads to sea quark pairs
¯ etc. (and gluons), in addition to the valence quarks.
ūu, dd
1.1.2
Neglection of electromagnetic interactions
One can compare the electromagnetic coupling constant (fine structure constant) to the
strong coupling parameter:
αem =
1
g2
e2
≈
αQCD =
= 0.1 · · · 1 ,
4π~c
137
4π~c
where the range of αQCD indicates that this parameter varies, depending on the process and
scales involved. This hierarchy suggests a simplification, the neglection of electromagnetic
interactions of the quarks.
1.1.3
Natural units
The above equation is written out in so-called Heaviside-Lorentz units. We will use natural
units1 ~ = c = 1 instead. This means that the units of distance and time and of energy and
mass become interchangeable (c = 1). Moreover, we do not distinguish between momentum
and energy or between momentum and inverse distance units (~ = 1).
1
In the modern North-American literature these are sometimes referred to as God-given units. The
connection between natural units and Nature is somewhat indirect, like that between a canonical basis and
a canon. In Nature there exist conservation laws that relate different units to each other. For instance
the speed of light is a constant and therefore distance can be measured (and defined) in terms of time.
Natural units are no inventions of Nature or supernature but they are one possible choice that we make.
Choosing these units leads to notational simplifications in quantum field theories, which is why this unit
system appears particularly “natural”, in this context.
3
• We quote momenta, energies and masses in GeV = 1000 MeV = 109 eV (units of
energy). This is approximately the mass of the proton mp .
• We measure distance and time in fm = 10−15 m. This is approximately the radius of
the proton rp .
• We can convert between GeV and fm:
0.197 fm · GeV ≈ 1 .
1.1.4
Hadron masses
We compare the masses of quarks, the nucleons and the pions:
• mu ≈ 2 MeV, md ≈ 5 MeV, ms ≈ 100 MeV,
• mp ≈ 938 MeV, mn ≈ 940 MeV,
• mπ± ≈ 140 MeV, mπ0 ≈ 135 MeV.
Note that atomic binding energies EB (i.e. the difference between the mass of an atom and
its constituent nucleus and electrons) are related to the radius of a state through the virial
theorem for the Coulomb potential as EB = −αem hr−1 i/2. The fact that mp ≈ EB ≈ 5 rp−1
(note the sign!) indicates that the QCD coupling is large and that something very different
from (Q)ED is going on. Why is the proton stable if its constituents are lighter than the
proton itself?
The masses of the hadrons are much bigger than the masses of their valence quarks. Moreover, md − mu ms , mπ , mn . So usually the approximation mu ≈ md ≈ mud = (mu + md )/2
is employed and electromagnetic interactions are neglected. In this particular limit the socalled isospin symmetry is realized and mπ± = mπ0 and mp = mn . To resolve the O(1 MeV)
corrections to this approximation, one would have to include the two sources of isospin
breaking (md − mu mass difference and the electric charge difference).
1.2
The QCD Lagrangian
The Lagrange density of QCD is given as
nf
X
1
µν
Ψ̄(x)f iDµ γ µ − mf Ψ(x)f .
LQCD (x) = − 2 tr Fµν (x)F (x) +
2g
f =1
(1)
This notation is very compact and we need
detail. The first part
Pnf to explain itµ in more
Ψ̄(x)f iDµ γ − mf Ψ(x)f contains the quark
contains the gluons (F). The second part f =1
and antiquark fields Ψf and Ψ̄f :
• f ∈ {u, d, s, . . .} is the flavour index and mf is the mass of a quark of flavour f .
4
• Dµ is the covariant derivative Dµ = ∂µ + iAµ with ∂µ = ∂/∂xµ . The Lorentz index
µ = 0, 1, 2, 3 labels the directions in space-time (0=time, 1,2,3=space)
P 2 −1 a
a
a
a
• Aµ (x) = N
a=1 Aµ (x)T is the four-vector potential with Aµ (x) ∈ R and T are the
generators of the SU(N ) group (N = 3 in QCD). They satisfy
a
T ,T
b
=i
2 −1
N
X
f abc T c ,
c=1
where f abc ∈ R are totally anti-symmetric structure constants. In the fundamental
representation, these T a are complex, traceless and hermitian N × N matrices with
components Tij . The indices i, j = 1, . . . , N are colour indices and the following normalization of the T a holds: tr T a T b = δ ab /2. The T a are generators of the group
since
!
2 −1
N
X
ωaT a
∀ U ∈ SU(N ) : ∃ ω a ∈ [0, 4π) so that U = exp i
a=1
with U U † = 1, det U = 1
• γ µ are Dirac matrices (4 × 4 matrices in four spacetime dimensions) which satisfy
{γ µ , γ ν } = γ µ γ ν + γ ν γ µ = 2η µν 14×4 ,
where (ηµν ) = (η µν ) = diag(1, −1, −1, −1) is the metric tensor. This metric
the
P3 defines
j j
µ
µ ν
0 0
product of two four-vectors a and b: a · b = aµ b = a b ηµν = a b − j=1 a b . Note
µ
that a0 = a0 , aj = −aj for j = 1, 2, 3. The components of γ µ can be written as γα,β
with Dirac or spinor indices α, β = 1, 2, 3, 4.
• The fermion field Ψf has 4 × N components (spin × color). We can write: Ψαi f , where
α is the spinor index and i is the color index. In QCD N = 3 and i=1,2,3.
The first term on the r.h.s. of (1),
1
tr
2g 2
Fµν (x)F µν (x), describes the dynamics of the gluons.
• g 2 = 4παQCD is the coupling “constant”.
• Fµν is the field strength tensor:
Fµν = ∂µ Aν − ∂ν Aµ + i[Aµ , Aν ] = −i[Dµ , Dν ] =
X
a
Fµν
Ta .
a
In non-Abelian theories like QCD the commutator [Aµ , Aν ] does not vanish. One easily
sees that
X
1 a µνa
b
tr Fµν F µν =
F µνa Fµν
tr T a T b = Fµν
F
,
2
µν,ab
where the summation over µ, ν and a is implicitly assumed on the r.h.s.
5
We can now write the Lagrangian in a less compact way (again the summation over repeated
indices is assumed):
nf
X
1 a
µ
µνa
LQCD (x) = − 2 Fµν (x)F (x) +
Ψ̄αj (x)f iDµ,jk γαβ
− mf δjk δαβ Ψβk (x)f ,
4g
f =1
where
(Dµ )jk = ∂µ δjk + iAaµ (T a )jk .
(2)
Dµ is spin-diagonal and γ µ is colour-diagonal.
It turns out that g 2 = g 2 (µ), mf = mf (µ) where µ is a momentum scale. As long as a physical
observable only depends on processes with momenta q µ, this should be independent of
the cut-off µ. This running of the coupling and quark masses is a common feature of quantum
field theories: not directly measurable parameters need to be adjusted as functions of a scale
µ at which QCD is evaluated, to keep physical observables (e.g. hadron masses) constant
and well defined. In QCD this is possible and no new terms need to be introduced as µ is
sent to infinity. QCD is a renormalizable quantum field theory.
One way of regularising QCD is to discretize it on a lattice with spacing a ∼ 1/µ. Removing
this cut-off (a → 0) then corresponds to taking the continuum limit. QCD is asymptotically
free: it turns out that at large scales q g 2 (q) → 0 and the quarks only weakly interact. Near
the continuum limit where µ ∼ a−1 is large (and q µ can be taken large), a connection
can be made with perturbation theory. At low momenta, i.e. at typical hadronic distance
scales, however, the coupling becomes strong and non-perturbative methods are required.
1.3
1.3.1
Lattice QCD
Technical trick: Euclidean time
It is convenient to employ a Euclidean metric with xj = xj M , x4 = it = ix0 M where the
superscript “M” denotes the Minkowski spacetime. The integral over x4 then runs along the
imaginary axis but can be rotated to an integration along the real axis if there are no poles
in the first and third sectors of the complex plane. This is the case under certain conditions
that are usually met in QCD.
P
The product of two Euclidean four-vectors reads: aµ bµ = aµ bµ = 4µ=1 aµ bµ The Euclidean
γ matrices satisfy the anti-commutation relations
{γ µ , γ ν } = 2δ µν 1 ,
†
†
γµ = γµ ,
µ ∗
where (γ µ )αβ = (γβα
) (∗ denotes the complex conjugate and † denotes the Hermitian
adjoint). The Euclidean action (spacetime integral of the Lagrangian) becomes:
Z
SQCD = d4 x LQCD (x) =
)
(
nf
Z
X
1
tr [Fµν (x)Fµν (x)] +
Ψ̄(x)f (Dµ γ µ + mf )Ψ(x)f ,
= d4 x
2g 2
f =1
6
M
where we have employed the convention SQCD = −SQCD
. The n point Green functions
(see below) contain the same information about the theory as the partition function (see
below) that is defined by the action. Green functions can be analytically continued from the
Euclidean to the original Minkowski metric. It turns out that many properties are identical
in both metrics, for example particle masses. These therefore need not be continued back
into Minkowskian spacetime.
1.3.2
Putting QCD on a lattice
• We introduce a Euclidean lattice Λ with NS3 · NT sites (points) and a lattice spacing a.
We do not know the physical value of a a priori. We first have to “measure” a physical
observable on the lattice and relate it to its experimental value. Therefore, we can
only “set the scale” a(g 2 , mf ) a posteriori. The lattice is a mass-independent scheme
and so one would usually use the same a(g 2 ) for all mf rather than a(g 2 , mf ). This
requires an interpolation or extrapolation to the quark mass values that are realized in
Nature, where the connection to the experimental value can be made. If the computer
tells us a proton mass number Mp ∈ R+ then we can obtain the lattice spacing from
the relation Mp (g 2 )a−1 (g 2 ) = mp where mp denotes the experimental value.
• Notation:
x = (xµ ) ∈ Λ ,
xµ = anµ
(3)
with space-components n1 , n2 , n3 ∈ {0, . . . , NS − 1} and a time component n4 ∈
{0, . . . , NT − 1}. We define vectors µ̂ in the direction µ ∈ {1, 2, 3, 4} of modulus
|µ̂| = a. Λ usually has toroidal boundary conditions: the neighbouring point in the
direction j of a point with coordinate nj = NS − 1 has nj = 0. The same convention
is realized in the Euclidean time (4-)direction.
• The transporter between two neighbouring lattice sites
Z x+µ̂
µ̂
0
0
Ux,µ = P exp i
dxµ Aµ (x ) ≈ eiAµ (x+ 2 )) ∈ SU(3)
(4)
x
†
is called a gauge link, link variable or link. Ux,µ = Ux+µ̂,−µ
is the link connecting the site
x with its neighbour (distance a) in the direction µ.P stands for “path ordering” which
is important to specify how the exponent of these non-commuting variables should be
evaluated.
• Quarks reside on the lattice sites: Ψx = Ψ(x).
• We have to discretize derivatives:
1
(Ψ̄∂µ Ψ)(x) = Ψ̄x Ψx+µ̂ − Ψx−µ̂ 1 + O(a2 )
2a
1
†
Ψx−µ̂ 1 + O(a2 )
(Ψ̄Dµ Ψ)(x) = Ψ̄x Ux,µ Ψx+µ̂ − Ux−µ̂,µ
2a
7
(5)
x + ν̂
Ux,ν
Ux,µ
x
Ψx
x + µ̂
Figure 2: Quarks and gluons on the lattice
• This lattice discretization enables a simple realization of local gauge transformations.
For gx ∈ SU(3):
†
g
Ux,µ → Ux,µ
= gx Ux,µ gx+µ̂
,
Ψx → Ψgx = gx Ψx ,
Ψ̄x → Ψ̄gx = Ψ̄x gx† .
• We find that Ψ̄Dµ Ψ is gauge invariant:
1 g g g
g†
Ψ̄x Ux,µ Ψx+µ̂ − Ux−µ̂,µ
Ψgx−µ̂
2a 1
†
†
†
†
Ψ̄x gx gx Ux,µ gx+µ̂ gx+µ̂ Ψx+µ̂ − Ψ̄x gx gx Ux−µ̂† ,µ gx−µ̂ gx−µ̂ Ψx−µ̂
=
|{z}
|{z}
2a
| {z }
| {z }
Ψ̄g Dµg Ψg =
=1
=1
=1
=1
= Ψ̄Dµ Ψ .
Lattice discretizations are not unique but in the continuum limit different discretizations should give the same results (universality).
• It is easy to see that traces over products of links along closed loops, so-called Wilson
loops, are gauge invariant too. The “minimal” Wilson loop is the plaquette (closed
loop of links along a square):
µ̂
ν̂
2
†
†
Ux,ν
= eia [Fµν (x+ 2 + 2 )]
Ux,µν = Ux,µ Ux+µ̂,ν Ux+ν̂,µ
with Fµν (x) = Fµν (x) + O(a) ,
where we used the definition (4) as well as the Taylor expansion and the simplified
Baker-Campbell-Hausdorff (=Glauber) relation
Ax+µ̂,ν = Ax,ν + µ̂∂µ Ax,ν + O(a2 ) ,
1
eA eB ≈ eA+B+ 2 [A,B] .
b
Note that the leading lattice corrections to Fµν
= 2 Re tr Fµν T b must be of order a2 ,
due to the invariance of the real part of the trace of the plaquette under a → −a
(⇒ µ̂ → −µ̂). Moreover, Fµν = −Fνµ .
8
†
Ux+ν̂,µ
†
Ux,ν
Ux+µ̂,ν
x
Ux,µ
Figure 3: A sketch of the plaquette.
• With this definition of the plaquette one finds:
X 1 X 2N X X 1
µ̂
2
ν̂
tr
1
−
cos(a
F
x
+
Re
tr
(
1
−
U
)
=
+
) =
µν
x,µν
2
2
g 2 x µ>ν N
g 2 µ,ν
x
Z
X
X 1 X
a→0
4 1
4
2
=
a
−→
d
x
tr
F
(x)F
(x)(1
+
O(a
))
tr Fµν (x)Fµν (x) ,
µν
µν
2
2
2g
2g
x
µ,ν
µ,ν
which is the gluonic part of the Euclidean continuum action.
• We define:
1
Retr (1 − Ux,µν ) ≡ S ≡ 1 − U ,
N
which is real and bounded from above.
Sx,µν ≡
1.3.3
The QCD action
We can now write down a lattice version of the QCD action
SQCD =
X X
2N X
S
+
Ψ̄x f Af [U ]x,y Ψy f
g 2 (a) f x,y∈Λ
| {z }
≡β
= −β
X
U + const. + Sf [U ] .
The inverse coupling β controls the lattice spacing a(β). Asymptotic freedom means that
the continuum limit is reached for β → ∞ (g 2 → 0). The constant term can be dropped as
it cancels from expectation values (see below).
Af [U ]x,y (colour and spin indices are suppressed) is a sparse diagonally dominant lattice
discretization of the Dirac operator. For Wilson-like discretizations this matrix depends on
a parameter κf that is related to the quark mass mf via
1
1
1
mf =
−
.
2a κf
κc
9
κf is called a hopping parameter, κc is the critical hopping parameter. The quark mass
vanishes at κf = κc = 18 + O(g 2 ). For κf < κc , mf > 0.
| {z }
>0
Remark: a · mf is proportional to the inverse condition number of Af .
We will keep the discussion general and therefore there is no need to specify the matrix A.
1.3.4
Expectation values
We omit sums and products over flavours. All formulae are written for the case of one quark
flavour of mass m. This can trivially be generalized to nf mass degenerate or non-degenerate
quarks.
The expectation value of an observable O, e.g. an n-point Green- (or correlation-) function,
can be evaluated via the path integral:
Z
1
[dU ] [dΨ̄][dΨ] O[U, Ψ, Ψ̄] e−S[U,Ψ,Ψ̄]
hOi =
|{z}
Z Q
x,µ
dUx,µ
S[U, Ψ, Ψ̄] = Sg [U ] + Sf [U, Ψ, Ψ̄] = β
X
S + Ψ̄A[U ]Ψ .
[dU ] is the so-called Haar measure and Z is the partition function. The normalization
condition h1i = 1 implies
Z
Z = [dU ][dΨ̄][dΨ] e−S[U,Ψ,Ψ̄] .
Quarks follow the so-called Fermi-statistics. This means that the temporal boundary conditions for these fields should be antisymmetric (which can be implemented within A). It
also implies that a state cannot be occupied by more than one quark. To realize the latter
requirement, the quark field components need to be represented by Grassmann numbers.
Excursion: some facts on Grassmann numbers
• Two Grassmann numbers θi anti-commute:
θi θj = −θj θi
In particular: (θi )2 = 0. This means that the Taylor expansion of a function of
Grassmann numbers f ({θi }) = a+βi θi +cij θi θj +· · ·+cθ1 θ2 . . . θN (where i = 1, . . . , N ,
N even, a, cij = −cji ∈ R, βi Grassmann numbers) does not contain any terms that
are quadratic in any of the variables θi . For quark fields, i will run over spacetime
position, colour and flavour.
• Grassmann numbers commute with regular numbers a ∈ R:
θi a = aθi
10
• They have rather simple integration rules:
dNθ = dθN dθN −1 . . . dθ1 ,
dθi dθj = −dθj dθi ,
dθi θj = −θj dθi
Z
Z
dθ 1 = 0
dθ θ = 1
√
• These rules, for complex Grassmann numbers ηj = (θj + iφj )/ 2, lead to:
" N
#
Z
X
[dη ∗ ][dη] exp −
ηi∗ Mij ηj = det M
i,j
and
Z
−1
[dη ∗ ][dη] ηm ηn∗ exp −η † M η = Mmn
det M
For our purpose we substitute
ηm → Ψx,α,i
ηn∗ → Ψ̄y,β,j
Mmn → A[U ]x,y,α,β,i,j .
Then on a fixed gauge field background U = {Ux,µ }:
Z
[dΨ̄][dΨ] e−Ψ̄A[U ]Ψ ∝ det A[U ] ,
R
[dΨ̄][dΨ] ΨX Ψ̄Y e−Ψ̄A[U ]Ψ
R
= A−1
hΨX Ψ̄Y i|U =
XY [U ] ,
[dΨ̄][dΨ] e−Ψ̄A[U ]Ψ
where X = (x, α, i), Y = (y, β, j) are multi-indices: we can obtain a quark two-point function
on a fixed gauge field background (= quark propagator) by inverting the Dirac matrix.
On the lattice there are NS3 NT distinct sites, e.g., 323 64 = 221 ≈ 2 · 106 . The fermion matrix
A has (12NS3 NT )2 components. Most of these vanish, due to the next-neighbour nature of
discretized derivatives. However, A−1 is not sparse anymore. Fortunately, as we have seen
above, the Fermions can be integrated out analytically since they only appear as bilinears.
Afterwards we are left with the gluonic degrees of freedom only. The gluonic action is highly
non-linear and cannot be integrated out analytically. The lattice contains 4NS3 NT links, each
of which are complex 3 × 3 matrices, with 8 independent real components. High dimensional
integrals are best solved employing importance sampling (Markov) Monte Carlo techniques.
(i)
The method is as follows. A gauge ensemble {Ui } = {Ux,µ }, i = 1, . . . , n ( = set of configurations Ui ) is generated. The standard way of doing this at present is the so-called
11
hybrid Monte Carlo algorithm. Each of the configurations is generated with a probability
−S [Ui ]
[dUi ] e eff
, where
Z
Z
−Seff [U ]
−Sg [U ]
e
=e
[dΨ][dΨ̄]e−Sf [U,Ψ,Ψ̄]
= e−Sg [U ] det A .
If the number of independent gauge configurations n is sufficiently large for the central limit
theorem to hold, we can approximate the expectation value
n
1
1X
O[Ui ] + O √
,
hOi =
n i=1
n
√
up to a statistical error that can be reduced like 1/ n, as n is increased. This intrinsic (unbiased) statistical error is the origin of the term “measurement”. On n gauge configuration
n measurements are taken and the average is calculated. Any Ψ-dependence of O[Ui ] can
be integrated out since all bilinears ΨX Ψ̄Y within O can be replaced by A−1 [Ui ]XY , as we
have seen above. We also remark that the above statistical interpretation requires a real,
bounded action and this is the main reason for our transformation to Euclidean spacetime.
2
2.1
Computation of a hadronic two-point function
Two-point Green function
The expectation value
C(t) = h0|O(t + t0 )O† (t0 )|0i = hO(t + t0 )O† (t0 )i
(6)
is an example of a two-point Green or correlation function. O† (t0 ) creates a state (e.g. a
pion) at time t0 out of the vacuum |0i and O(t + t0 ) destroys it at later time t + t0 . The
translational invariance of expectation values
hO(x)i = hO(0)i =
1 X
hO(x)i
NS3 NT x∈Λ
allows us to choose t0 = 0. We could also average over different t0 -values, thereby reducing
the statistical error of the expectation value. However, this would also be more expensive,
computationally.
The evolution of a quantum mechanical state in Euclidean time is given by
|ψ(t)i = e−Ht |ψ(0)i ,
with a Hamiltonian H (that we do not need to know explicitly). For energy eigenstates
|ni this reads, e−Ht |ni = e−En t |ni, where En is the energy of the state |ni. We now set
12
|ψ(0)i = O† (0)|0i = Ô† |0i and insert a complete set
P
n
|nihn| = 1 into (6). This results in
C(t) = h0|O(t)O† (0)|0i = h0|Ôe−Ht Ô† |0i =
X
=
h0|Ô|nie−En t hn|Ô† |0i =
n
=
X
|h0|Ô|ni|2 e−En t =
X
n
|Cn |2 e−En t :
n
the correlation function is a sum of exponentials. Ordering E1 < E2 < · · · , we can write for
large enough times t:
X
|C2 |2 −(E2 −E1 )t
2 −En t
2 −E1 t
1+
C(t) =
|Cn | e
≈ |C1 | e
e
.
|C1 |2
n
E1 is the ground state energy. The combination E1 t = (E1 a)(t/a) = aE1 n4 (see Eq. (3)) is
dimensionless. On the computer we will obtain a dimensionless number aE1 ∈ R+ .
We define the effective mass as
C(t)
d
1
≈ − ln C(t)
Eeff (t) = ln
a
C(t + a)
dt
2
|C2 | −(E2 −E1 )t
= E1 + (E2 − E1 )
e
.
|C1 |2
|
{z
}
>0
At large times t Eeff will approach E1 , up to exponential corrections. If the same operator is
used at the source (t0 = 0) and the sink (t) then Eeff (t) will monotonously decrease towards
E1 (within statistical errors). The “speed” of this convergence is determined by the overlap
coefficients Ci . If O is chosen such that C1 Ci for i > 1 then the convergence will be fast.
2.2
Technical tricks for computing hadron masses
Trick 1 (Wick contractions): BXY is some arbitrary function of the spacetime-spin-colour
multi-indices X = (x, α, i) and Y = (y, β, j) that may also depend on the gauge configuration
U . We can cast the fermionic average (at fixed U ) of a quark bilinear into a trace of the
product of ordinary matrices:
X
X
hΨ̄x By,y Ψy i|U =
−htr Bxy Ψx Ψ̄y i|U
| {z }
x,y
x,y
=−
X
A−1
yx
tr Bx,y A−1
yx [U ] .
x,y
The trace is over spin and colour and we have kept spin and colour explicit: Axy and Bxy
are 12 × 12 matrices and Ψx is a twelve-component vector. Note the minus sign that originates from interchanging the two Grassmann fields. This trick can easily be generalized to
13
more complicated expectation values, where all combinations of quark-antiquark pairs (of
the same flavour) need to be contracted into A−1 factors.
Trick 2 (Dirac algebra):
γ5 ≡ γ1 γ2 γ3 γ4 = (γ4 γ3 γ2 γ1 )† = γ5† ,
γ5 γµ = −γµ γ5 ,
γ52 = 1 ,
γ5 γµ γ5 = −γµ
⇒ γ5 Dµ γµ γ5 = −Dµ γµ = Dµ† γµ = (Dµ γµ )†
⇒ γ5 Aγ5 = A† ,
γ5 A−1 γ5 = A†
−1
.
This last property is known as γ5 -Hermiticity. Also notice that γ5 A is Hermitian.
2.3
The pion two-point function
A pion π with spatial momentum p can be destroyed by the operator
X
(f1 ) 5
(f2 )
e−ipx Ψ̄x,α,j
γα,β Ψx,β,j
,
x,αβ,j
where the superscripts (f1 ) and (f2 ) denote the flavour, in this case up and down quark.
The continuum dispersion relation reads E12 = m2π + p2 , where E1 is the ground state energy
and mπ the ground state mass. Up to O(a) (or O(a2 ), depending on the fermionic action)
corrections this also holds on the lattice. For notational simplicity, we restrict ourselves to
rest frame i.e., to momentum p = 0 ⇒ E1 = mπ .
A general mesonic destruction operator (or interpolator) of quark-antiquark type can be
written as
X
e−ipx Ψ̄(f1 ) ΓD[U ]Ψ(f2 ) x ,
x
where we have dropped the colour and spin indices. Γ denotes a product of Γ-matrices and
D[U ] a gauge covariant spin-diagonal matrix that may contain discretized covariant derivatives (to realize angular momentum) and/or a smearing function, to enhance the projection
onto the ground state.
We define Ψ̄ ≡ Ψ† γ4 . This is the Minkowski definition. In Euclidean spacetime the substitution Ψ̄ → Ψ† in all definitions would be more consistent. However, this allows us to copy
all operators one-to-one from the literature that mostly employs Minkowski conventions. As
a warm up we perform a manipulation needed to obtain the creation operator:
Ψ̄(f1 ) ΓΨ(f2 )
†
= Ψ(f1 ) † γ4 ΓΨ(f2 )
†
= −Ψ(f2 ) † Γ† γ4 Ψ(f1 )
= −Ψ(f2 ) † γ4 γ4 Γ† γ4 Ψ(f1 ) = ∓Ψ̄(f2 ) ΓΨ(f1 ) ,
depending on γ4 Γ† γ4 = ±Γ. (For Γ = γ5 : γ4 γ5 γ4 = −γ5 the + sign is correct.)
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Now we can write down an expression for the pion two-point function.
X
(f2 )
(f2 )
1)
1)
Cπ (t) =
Ψ̄(f
Ψ̄y γ5 Ψ(f
, x4 = t , y4 = 0
x γ5 Ψx
y
x,y
= −NS3
XD
h iE
(f2 )
(f1 ) (f1 )
2)
tr γ5 Ψ(f
Ψ̄
γ
Ψ
Ψ̄
5
0
0
x
x
x
Trick 1
= −NS3
X
−1
tr γ5 A−1
x0 γ5 A0x
x
Trick 2
=
−NS3
XD
tr
(A−1 )†0x A−1
0x
E
,
x
where we assume flavours f1 and f2 to be mass-degenerate and, therefore, do not distinguish
between A(f1 ) and A(f2 ) . Going from the first to the second line, the translational invariance
of expectation values was exploited and the minus sign stems from commuting the Grassmann numbers. We remark that in the case of equal flavours f1 = f2 (so-called flavour
singlets), a second pairing, with a relative minus-sign would have appeared, applying the
−1
Wick contraction (trick 1). This additional term contains A−1
00 and Axx , that are harder to
compute. In the non-flavour singlet case above, only 12 rows of A−1 are required.
The A−1
0x are so-called point-to-all propagators. These are sets of 12 Dirac-vectors (one
for each source spin-colour combination), with 12NS3 NT elements each. Now imagine we
−1
would have needed A−1
yx for all spacetime positions y! How do we compute A0x ? We define
point-sources:
0 β j
b(β,j)
z,γ,κ = δz δγ δκ
and solve the 12 sparse linear systems
As(β,j) = b(β,j) .
This gives the point-to-all propagator:
(β,j)
−1
(β,j)
sx,α,i = A−1
x,α,i,z,γ,κ bz,γ,κ = Ax,α,i,0,β,j .
Remark: A is not Hermitian. It is also possible to solve the system A† As = A† b, instead,
where A† A = (γ5 A)(γ5 A) is Hermitian. Naively, this requires additional multiplications.
Once the 12 solutions are computed, we can obtain the correlation function
X
(β,j)
Cπ (t) = −NS3
h|sx,α,i |2 i
(7)
~
x,α,i,β,j
and extract the mass from its large t behaviour. Smearing may be implemented in the
operator too, to enhance the overlap with the physical ground state. In this case, instead of
the point sources above one would use smeared sources. We have claimed that the correlation
function should be positive. This is not the case above. This can be traced back to the fact
that we have not consistently implemented Euclidean conventions (Ψ̄ instead of Ψ† ). Some
15
software packages like Chroma drop in an extra minus sign that leads to positive pion and
rho correlation functions but then for instance the scalar becomes negative and this overall
sign needs to be corrected by hand.
Finally, it is easy to see that the result for a meson with general Γ (Ψ̄(f1 ) ΓΨ(f2 ) ) reads
X
(f1 )(β,j) ∗
(f2 )(δ,j)
C(t) = ±
sx,α,i
(γ5 Γ)αγ sx,γ,i
(Γγ5 )δβ .
x,α,β,γ,δ,i,j
(Note that for the pion Γ = γ5 and γ5 γ5 = 1) Since in standard notations each γ5 Γ has only
four non-vanishing components, computer time can be saved by tabulating the relevant spin
bilinears.
3
Further reading
Grassmann variables and Wick rotation
• J. Smit, Introduction to quantum fields on a lattice: A robust mate, Cambridge Lect.
Notes Phys. 15 (2002) 1.
Computation of 2-point functions and Wick contractions
• C. Gattringer and C. B. Lang, Quantum chromodynamics on the lattice, Lect. Notes
Phys. 788 (2010) 1.
• I. Montvay and G. Munster, Quantum fields on a lattice, Cambridge, UK: Univ. Pr.
(1994) 491 p.
Fermion actions and fermion doubling
• M. Creutz, Quarks, Gluons And Lattices, Cambridge, UK: Univ. Pr. (1983) 169 p.
• T. DeGrand and C. E. Detar, Lattice methods for quantum chromodynamics, New
Jersey, USA: World Scientific (2006) 345 p.
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